Nicholas Williams
I am a Postdoctoral Research Associate at the Lancaster University working with David Pauksztello, where I have been since the beginning of September 2022.
I received my doctorate from the University of Cologne in 2022 under the supervision of Sibylle Schroll.
Following that, I was a JSPS short-term postdoctoral research fellow at the University of Tokyo working with Osamu Iyama from late May to late August 2022.
My research focuses on the representation theory of finite-dimensional algebras and combinatorics.
In the representation theory of finite-dimensional algebras I am particularly interested in
- higher homological algebra,
- maximal green sequences,
- stability conditions,
- relations with cluster algebras.
- partially ordered sets, especially the higher Bruhat and Stasheff–Tamari orders,
- polytopes and their triangulations, especially cyclic polytopes.
You can find a copy of my CV here (updated 28/11/22).
GitHub,
ResearchGate,
Google Scholar,
ORCID
Preprints
| 8/2022 | Stability spaces of string and band modules, (with Sibylle Schroll, Aran Tattar, Hipolito Treffinger, and Yadira Valdivieso) arXiv. |
| 8/2022 | Triangulations of prisms and preprojective algebras of type A, (with Osamu Iyama) arXiv. |
| 06/2021 | The two higher Stasheff–Tamari orders are equal, arXiv. |
| 12/2020 | The first higher Stasheff–Tamari orders are quotients of the higher Bruhat orders, arXiv. |
Journal publications
| To appear | Quiver combinatorics and triangulations of cyclic polytopes |
| Algebraic Combinatorics. arXiv. | |
| 10/2022 | New interpretations of the higher Stasheff–Tamari orders |
| Advances in Mathematics. DOI, arXiv. | |
| 9/2021 | The combinatorics of tensor products of higher Auslander algebras of type A, (with Jordan McMahon) |
| Glasgow Mathematical Journal. DOI, arXiv. |
Conference proceedings
| To appear | The higher Stasheff–Tamari orders in representation theory |
| Proceedings of the 2020 International Conference on Representations of Algebras. | |
| 2021 | The first higher Stasheff–Tamari orders are quotients of the higher Bruhat orders |
| Proceedings of the 33rd International Conference on Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire 85B (2021), Art. 6, 12. Link. |
Other publications
| 7/2022 | Microthesis: Triangulations of cyclic polytopes in representation theory |
| London Mathematical Society Newsletter, No. 501, 33-34. Link. | |
| 4/2022 | Higher-dimensional combinatorics in representation theory |
| Doctoral thesis, University of Cologne. Link. |
Recent talks
| 11/2022 | The higher Stasheff–Tamari orders on triangulations of cyclic polytopes |
| Séminaire du LaCIM, Université du Québec à Montréal | |
| 11/2022 | Cyclic polytopes and representation theory |
| Geometry and mathematical physics seminar, University of Birmingham | |
| 11/2022 | Cyclic polytopes and representation theory |
| Pure mathematics seminar, Lancaster University | |
| 10/2022 | Mutation in higher cluster categories |
| Seminar@Aarhus, Aarhus University | |
| 9/2022 | A structural approach to maximal green sequences |
| Silting theory, algebras and representations, Charles University, Prague | |
| 7/2022 | Algebraic interpretation of the higher Stasheff–Tamari orders |
| Representation theory and various issues on related topics, Research Institute for Mathematical Sciences, Kyoto University | |
| 7/2022 | Cyclic polytopes and higher Auslander–Reiten theory III |
| Tokyo–Nagoya algebra seminar, University of Tokyo and Nagoya University | |
| 6/2022 | Cyclic polytopes and higher Auslander–Reiten theory II |
| Tokyo–Nagoya algebra seminar, University of Tokyo and Nagoya University | |
| 6/2022 | Mutating cluster-tilting objects in (d + 2)-angulated cluster categories |
| FD seminar | |
| 6/2022 | Cyclic polytopes and higher Auslander–Reiten theory I |
| Tokyo–Nagoya algebra seminar, University of Tokyo and Nagoya University | |
| 2/2022 | Equivalence of maximal green sequences |
| Paris algebra seminar, Institut de Mathématiques de Jussieu-Paris Rive Gauche | |
| 1/2022 | The first higher Stasheff–Tamari orders are quotients of the higher Bruhat orders |
| Formal Power Series and Algebraic Combinatorics | |
| 1/2022 | An algebraic interpretation of the higher Stasheff–Tamari orders |
| Aachen–Bochum–Cologne Darstellungstheorie Seminar | |
| 12/2021 | Cyclic polytopes and higher Auslander–Reiten theory |
| Tokyo–Nagoya algebra seminar | |
| 4/2021 | The higher Stasheff–Tamari orders in representation theory |
| Paris algebra seminar, Institut de Mathématiques de Jussieu-Paris Rive Gauche | |
| 3/2021 | New interpretations of the higher Stasheff–Tamari orders |
| Diskrete Mathematik / Geometrie Forschungsseminar, TU Berlin | |
| 3/2021 | The Tamari lattice in representation theory |
| Junior Algebra Colloquium, University of Bristol |
Translations
I have translated a couple of papers of Carathéodory from German into English. I make no claim to any of the mathematics in these papers.
It is also not impossible that I have introduced errors in the process of translation.
| Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen |
| Constantin Carathéodory |
| Mathematische Annalen (1907), Volume: 64, pp. 95–115. |
| Original, Translation |
| Über den Variabilitätsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen |
| Constantin Carathéodory |
| Rendiconti del Circolo Matematico di Palermo (1911), Volume: 32, pp. 193–217. |
| Original, Translation |
Categorifications in Representation Theory
I organised this conference with Aran Tattar.
Other
I am sometimes asked what the painting behind me on zoom is. It is one of my uncle's paintings. You can find more of his paintings here.
